Identify the wave number from the given wave equation.

The general equation for a progressive wave is $y = A\sin(\omega t - kx)$, where $k$ is the wave number.
The given equation is $y = 0.1\sin (400\pi t - 20\pi x)$.
By comparing the two equations, we can see that the wave number $k = 20\pi \text{ rad/m}$.

The phase difference $\Delta\phi$ between two points separated by a distance $\Delta x$ is given by the formula:

We are given $\Delta x = 0.05 \text{ m}$.

The calculated phase difference is $\pi$ radians, which corresponds to option B.

The correct option is B. $\boxed{\text{B. } \pi \text{ radians}}$